Mixed Time-frequency Domain Research Articles

Geometric Algebra Applied to Multiphase Electrical Circuits in Mixed Time–Frequency Domain by Means of Hypercomplex Hilbert Transform

In this paper, power flows in electrical circuits are modelled in a mixed time-frequency domain by using geometric algebra and the Hilbert transform for the first time. The use of this mathematical framework overcomes some of the limitations of some of the existing methodologies, in which the so-called “active current” may not lead to the lowest Root Mean Square (RMS) current under distorted supply or unbalanced load. Moreover, this current may contain higher levels of harmonic distortion compared to the supply voltage. The proposed method can be used for sinusoidal and non-sinusoidal power supplies, non-linear loads and single- and multi-phase electrical circuits, and it provides meaningful engineering results with a compact formulation. It can also serve as an advanced tool for developing algorithms in the power electronics field. Several examples have been included to verify the validity of the proposed theory.

Large Area Detection of Microstructural Defects with Multi-Mode Ultrasonic Signals

Cyclic loading or other stresses can lead to development of cracks and crack growth in mechanical structures, leading to eventual failure. While ultrasound imaging can be used for non-destructive testing of such structures, conventional ultrasound techniques are often limited by crack size, density, and areal coverage. An effective characterization of real-world, large-area structures is required at an early damage stage to prevent catastrophic failure and predict remaining life. In this study, a new nonlinear ultrasonic testing (NUT) method is proposed for large-area monitoring of practical structures with arbitrary complexity by using multiple-mode guided-wave ultrasonic signals. The proposed guided-wave NUT technique requires single-element transducers, simple electronics, and a mixed time-frequency domain signal processing. As a proof-of-concept demonstration, numerical simulations and experiments are performed on an A36 carbon steel beam assembly with previously formed microstructural defects that cause nonlinearities in ultrasonic response. The quadratic dependence of the nonlinear wave excitation on the input ultrasonic signal amplitude is shown by numerical simulations, and such a nonlinear ultrasonic response is experimentally observed in the zone with a high density of microstructural defects.

Iterative Accurate Seismic Data Deblending by ASB-Based Robust Sparse Radon Transform

Blended acquisition can improve the acquisition efficiency, and thereby reduce the acquisition costs. It becomes an important acquisition method in seismic exploration. However, the blending noise imposes challenges for subsequent traditional seismic data processing procedures, and thus deblending algorithms are necessary to obtain deblended seismic data. Based on the assumption that the signal is coherent and the blending noise is randomized in a specific domain, traditional deblending methods using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -norm measuring the data misfit can obtain separated gathers. However, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -norm is improper when the appearing seismic noise bias the normal distribution, that is, abnormal noise appears. To attenuate the abnormal noise effects during iterative deblending, we proposed an accurate deblending algorithm based on a robust sparse Radon transform (RSRT). For the RSRT, the alternating split Bregman (ASB) algorithm is used for robust 2-D Radon model updating with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> -norm to measure data misfit in the mixed time–frequency domain and the sparsity constraint to the time-domain Radon model. Using the RSRT iteratively, the Radon model and the corresponding deblended data can be estimated robustly, accurately, and efficiently. Blended synthetic data with different levels of abnormal noise and with different trace intervals demonstrate the validity and flexibility of the proposed robust deblending method quantitatively with a high recovered SNR. Numerically blended field data further prove the effectiveness of the proposed method in attenuating the abnormal and blending noise.

A Robust and Efficient Sparse Time-Invariant Radon Transform in the Mixed Time–Frequency Domain

The Radon transform (RT) has been widely used as a powerful tool, especially in exploration geophysics fields, such as multiple removal, interpolation, and velocity analysis. However, the existing strong outlier effects can seriously decrease the accuracy of the traditional RT. Therefore, a robust time-invariant RT (TIRT) is proposed in the mixed time–frequency domain to attenuate the outlier effects by using double L1-norm sparse constraints performed on the data misfit and the Radon model in the time domain. For the TIRT, the forward RT and its adjoint can be implemented in the frequency domain efficiently. Only one matrix inversion for each frequency component is involved in all iterations to speed up the iterations. Then, the 1-D alternating split Bregman (ASB) algorithm is introduced and improved for 2-D Radon model updating efficiently. It involves matrix-vector multiplication operators and two proximity operators. These two proximity operators can guarantee the robustness and sparseness of the proposed method. Numerical examples of synthetic and field data demonstrate the effectiveness and validity of the proposed method. The proposed method is also used for interpolation to decrease the trace interval. After interpolation, seismic data are more continuous with less serrations along the spatial direction and the frequency-wavenumber spectrum is more focused. The interpolated data have wider potential applications in improving the accuracy of the following seismic processing. It should be noted that the proposed robust and efficient RT can also be used in remote sensing and computerized tomography fields instead of the traditional RT.

A method for numerical and experimental nonlinear modal analysis of nonsmooth systems

Abstract The development of nonlinear modal analysis so far has focused on structures with smooth nonlinearities. However, nonsmooth nonlinearities, which are, for instance, caused by contact interactions are highly relevant in practical applications. This paper proposes a novel numerical approach along with a method for the measurement of nonlinear modes of structures with nonsmooth contact nonlinearities. The proposed numerical method combines the shooting method and the harmonic balance method, yielding a mixed time-frequency domain representation of the system, allowing for an efficient treatment of the nonsmooth contact law within the numerical approach. Moreover, the mass of the system is redistributed such that the contact nodes are massless. Thereby, the dynamic contact problem can be reduced to a quasi-static contact problem. A salient feature of this numerical approach is that the contact problems are solved without the need for any contact parameters, such as penalty or restitution coefficients. Furthermore, the conservative nature of the contact law incorporated in this formulation allows for the calculation of nonlinear modes as periodic solutions of conservative systems. The experimental method relies on a nonlinear phase resonance approach. Hitherto, phase resonance methods have exclusively been applied to systems with smooth nonlinearities. In this study, an automated nonlinear phase resonance approach with phase-controlled excitation is used, providing a robust experimental procedure, which facilitates the treatment of strong nonsmooth nonlinearities, e.g., caused by unilateral constraints inducing impacts. The numerical and experimental methods are demonstrated by an application to a benchmark structure consisting of a beam with one-sided support leading to impacts. It is shown that the numerical method can be applied without the need for any nonlinear system identification effort and the results agree well with the measured nonlinear modes.

Capturing Structural Snapshots during Photochemical Reactions with Ultrafast Raman Spectroscopy: From Materials Transformation to Biosensor Responses.

Chemistry studies the composition, structure, properties, and transformation of matter. A mechanistic understanding of the pertinent processes is required to translate fundamental knowledge into practical applications. The current development of ultrafast Raman as a powerful time-resolved vibrational technique, particularly femtosecond stimulated Raman spectroscopy (FSRS), has shed light on the structure-energy-function relationships of various photosensitive systems. This Perspective reviews recent work incorporating optical innovations, including the broad-band up-converted multicolor array (BUMA) into a tunable FSRS setup, and demonstrates its resolving power to watch metal speciation and photolysis, leading to high-quality thin films, and fluorescence modulation of chimeric protein biosensors for calcium ion imaging. We discuss advantages of performing FSRS in the mixed time-frequency domain and present strategies to delineate mechanisms by tracking low-frequency modes and systematically modifying chemical structures with specific functional groups. These unique insights at the chemical-bond level have started to enable the rational design and precise control of functional molecular machines in optical, materials, energy, and life sciences.

A Dual Monitoring Technique to Detect Power Quality Transients Based on the Fourth-Order Spectrogram

This paper presents a higher-order statistics-based approach of detecting transients that uses the fourth-order discrete spectrogram to monitor the power supply in a node of the domestic smart grid. Taking advantage of the mixed time–frequency domain information, the method allows for the transient detection and the subsequent identification of the potential area in which the fault takes place. The proposed method is evaluated through real power-line signals from the Spanish electrical grid. Thanks to the peakedness enhancement capability of the higher-order spectra, the results show that the procedure is able to detect low-level transients, which are likely ignored by the traditional detection procedures, where the concern pertains to power reliability (not oriented to micro grids), and this, by analyzing the duration and frequency content of the electrical perturbation, may indicate prospective faulty states of elements in a grid. Easy to implement in a hand-held instrument, the computational strategy has a 5 Hz resolution in the range 0–500 Hz and a 50 Hz resolution in the range of 0–5 kHz, and could be consequently used by technicians in order to allocate new types of transients originated by the distributed energy resources. Four real-life case-studies illustrate the performance.

Selective Determination of Floquet Quantities for the Efficient Assessment of Limit Cycle Stability and Oscillator Noise

We present a mixed time-frequency domain algorithm for the approximate computation of the Floquet quantities (exponents and both direct and adjoint eigenvectors) resulting from the linearization of index-1 differential-algebraic equations around a periodic limit cycle. The approach allows to select the number of Floquet exponents to be calculated approximating the matrix of the obtained eigenvalue problem. The error in the evaluation (for both exponents and eigenvectors) is proved to tend to zero along with the ratio between the norms of the neglected and retained rows of the relevant matrix.

Observer/Kalman Filter Identification With Wavelets

Understanding the dynamic characteristics of the target system is a fundamentally important step in designing reliable closed loop control systems. One method for identifying linear models from the target uses the observer/Kalman filter identification/eigensystem realization algorithm (OKID/ERA) combination. This focuses on time domain representations of plant input and output data. The wavelet transform is capable of providing an efficient mixed time-frequency domain representation of time domain data. We have investigated if this would give any benefit for OKID/ERA. We show how to apply the wavelet transform within the OKID creating a new wavelet-OKID/ERA technique and then compare its performance, using common discrete wavelet families, against a standard procedure. Results indicate that the potential attractiveness of the wavelet approach do not translate into a practical reality.

Mixed time–frequency domain simulation of a hydraulic inductance pipe with a check valve

This paper deals with the efficient computation of hydraulic switching systems with a check valve by a mixed time–frequency domain method; frequency-domain modelling is performed on the wave propagation in a pipe and time-domain modelling is applied to the switching valve and the check valve. The dual property of the check valve makes the complete problem have variational inequality properties. A solution method is presented which replaces the pressure and the flowrate of the check valve as a function of one new variable. The resulting system of non-linear algebraic equations is solved using a Newton–Raphson method in combination with a smoothing of the non-smooth properties of the check valve. The method is applied to a parameter study of a hydraulic buck converter.

A Linearized Small-Signal Model of an HVDC Converter With Filter Circuits in Mixed Time-Frequency Domain

In this paper, a linearized small-signal model for a six-pulse HVDC system is presented. In this paper, the effects of on and off states resistances, forward voltage drop, and snubber circuits of the thyristors and filter circuits are taken into account. RL series circuits are used for modeling of the converter transformer and ac system. To validate the model developed in this paper, a comparison is made with the PSCAD/EMTDC time-domain simulation program.

Covariance NMR spectroscopy by singular value decomposition

Covariance NMR is demonstrated for homonuclear 2D NMR data collected using the hypercomplex and TPPI methods. Absorption mode 2D spectra are obtained by application of the square-root operation to the covariance matrices. The resulting spectra closely resemble the 2D Fourier transformation spectra, except that they are fully symmetric with the spectral resolution along both dimensions determined by the favorable resolution achievable along ω 2. An efficient method is introduced for the calculation of the square root of the covariance spectrum by applying a singular value decomposition (SVD) directly to the mixed time-frequency domain data matrix. Applications are shown for 2D NOESY and 2QF-COSY data sets and computational benchmarks are given for data matrix dimensions typically encountered in practice. The SVD implementation makes covariance NMR amenable to routine applications.

Time–frequency theory of pump-probe absorption spectroscopy

A variation of density matrix formulation based on the nature of field–matter interference in a mixed time–frequency domain is developed to study molecular pump-probe absorption spectra in condensed phases. Considered are both the integrated probe transmitant signals and the frequency-dispersed transient absorption coefficients for molecular systems with either two or three electronic surfaces involved. The present formulation is exact and applicable to any field with arbitrary timescale and shape, and it is valid when the pulses are overlapped as well as separated. The dual temporal-spectral effect of both the excitation and detection fields can be clearly elucidated via a transformation that preserves causality. The resulting field–matter interference picture of transient absorption in the correlated time-frequency domain is conceptually natural and physically transparent. The molecular dynamics, and the field–matter temporal/spectral coherence and interference phenomena can all be clearly demonstrated.

Light-matter temporal and frequency coherence and interference

The effects of coherence and interference in light-matter systems are treated using a novel causality preserving density matrix formulation that operates in a mixed time-frequency domain and applies to quantum as well as classical systems. Within the limits of the classical weak field and the rotating wave approximations, this formulation can be applied to arbitrary light fields, including incoherent light, transform limited ultrashort light pulses, chirped pulses, and phase-locked pulses. The light field function and the response function of the material system can be separately evaluated, and then convoluted in time and frequency to give the optical signal.

The Wigner distribution of linear time-variant systems

The Wigner distribution relation between the two input and output time-varying spectra of the linear time-variant system is derived. This mathematical relationship can show how a time-variant system modifies the time-varying spectrum of the input to determine that of the output, and this result will be very useful for time-variant filtering in the mixed time-frequency domain. A linear time-variant digital filtering example is used to show the effect of time-variant filtering by using the Wigner distribution. >

Time-variant filtering of signals in the mixed time frequency domain

A new approach to the problem of time-variant filtering is presented. This approach is based upon the generation of the mixed time-frequency representation (MTFR) of a signal, multiplication of that representation by a time-frequency function H(ω,t), and obtaining a filtered output by an inverse operation. The resultant filter is linear if the time-frequency representation used is the complex spectrogram. In contrast, the filter is nonlinear if the Wigner distribution function is used. Not every function of two variables is an allowed MTFR of a signal; some conditions must be satisfied. If the function produced by the product of the signal MTFR and the filter function is not an allowed MTFR, an approximation based on projection onto the space of allowed MTFR functions is investigated. This approximation yields a filtered function whose MTFR is as close as possible (in the least-squared sense) to the desired.

Optical time-variant processing of signals in the mixed time-frequency domain

A new approach to the problem of time-variant filtering is presented. This approach is based upon the generation of the mixed time-frequency representation of a signal, multiplication of that representation by a 2-D function H(ω, t), and obtaining a filtered output by an inverse operation. An astigmatic optical system is used to implement the above operations. The resultant filter is linear if the time-frequency representation used is the complex spectrogram. In contrast, the filter is non-linear if the Wigner distribution function is used.

Response of Linear Periodically Time Varying Systems to Random Excitation

R ECENTLY developed stochastic models for the flapping vibration of helicopter rotor blades give rise to the problem of determining the response of periodically time varying linear systems to random excitation. Mathematically, the problem is to determine the statistical properties of a response process x(t) satisfying dx/dt = A(t)x + /; A(t) = A(t + T) (1) where/(t) is a random input process and the matrixv4(£) is given in terms of system parameters. In this Note, the class of weakly periodic nonstationary processes is shown to arise in a natural way in connection with the system (1). In particular, it is shown that the steady-state response of the system (1), if it exists, to a weakly periodic nonstationary process with period T is also a weakly periodic nonstationary process with period T. In Refs. 1-4 both time domain and mixed time frequency domain methods have been used to calculate response statistics for Eqs. (1). These relations are generalized within the framework of weakly periodic nonstationary processes.